Hamana's injective envelope as a maximal rigid multiplier cover

Abstract

Let $A$ be a unital $C^*$-algebra. We call an $A$-multiplier cover a pair $(E,\iota)$ consisting of a $C^*$-algebra $E$ and a faithful non-degenerate $*$-homomorphism $\iota\colon A\to M(E)$. Ordering such covers by $A$-preserving unital completely positive maps between multiplier algebras, we study those covers for which the inclusion $A\subseteq M(E)$ is rigid in Hamana's sense. We prove that Hamana's injective envelope $I(A)$ is a maximal rigid $A$-multiplier cover and that, conversely, a rigid cover is maximal if and only if its multiplier algebra is canonically $*$-isomorphic to $I(A)$ over $A$. Thus maximal rigid multiplier covers provide an order-theoretic characterisation of the injective envelope. In the commutative case $A=C(X)$, this recovers the familiar realisation $I(C(X))\cong C(G(X))\cong M(C_0(U))$ for a dense cozero set $U$ in the Gleason cover $G(X)$, in a form inspired by B{\l}aszczyk's concise construction of the Gleason cover.